You are playing a game where there are 9 boxes laid out in a row, numbered 1 through 9 from left to right. Randomly placed in one of the boxes is a slip of paper that says "GAME OVER". The other eight boxes each contain $1000.

You are to pick the boxes one at a time. If you pick a box with $1000, you keep the money and you must pick another box. If at any point you select the box that says "GAME OVER", the game ends and you leave with the prize money you've accumulated to that point. The only catch is, the host of the game show will tell you which direction the "GAME OVER" box is in, and you must guess your next box in that direction (it's like a guessing game where you have to guess a number from 1 to 9, and after each guess the host tells you "higher" or "lower" until you finally guess the number he is thinking of). However, the goal of this game is not to land on the "GAME OVER" box (since you eventually will), but to maximize the number of guesses you take (and thus your profit) before you land on it.

Question 1: Is there an optimal strategy for this game? If so, what is it and what is your expected profit? If not, why not?

Question 2: What if you were the host, and instead of randomly placing the "GAME OVER" box, you could choose where it went - is there a strategy that would minimize the expected profit of the contestant?

There are two different ways to interpret the game. In one way, the player cannot go back to get other values, for instance, if the player chooses 5 and the host says "left", the boxes 6-9 are forever eliminated for that game. The other way, the player can go back and retrieve other boxes.

In the no-go-back way, the best method for choosing boxes is an alternating choosing scheme starting with either 1 or 9. By conducting 200 trials with randomly selected numbers, we found the estimated value for this method was much higher than the rest, giving a value in the high 4000's range.

For the can-go-back way, the best method would be to start at 5, and then choose either 1 or 9, depending on which direction the host says to go. Then go and get the remaining boxes on the other side of 5. This will guarantee the player at least 6000, if the game over is not in 5,1, or 9.

Answering question 2, the best box that the host could assign the "game over" slip to would be either box 1 or 9. If you take the best strategies from either of the two games above, then you will see that 1 or 9 is always an early pick, however, even if not chosen first, the estimated value of prize money possible for the numbers 1 or 9 was $1714.29. This was taken using randomly selected numbers and seven different strategies such as, both alternating methods, starting at the ends and working across, starting at 5 and going back to get values, and starting at 4 and going back to get values.